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N O N L I N
Nonlinear Regression Analysis Program
Phillip H. Sherrod
A "shareware" program
Nonlin allows you to perform statistical regression
analyses to estimate the values of parameters for
linear, multivariate, polynomial, and general nonlinear
functions. The regression analysis determines the
values of the parameters which cause the function to
best fit the observed data that you provide. Nonlin
allows you to specify the function whose parameters are
being estimated using ordinary algebraic notation. In
addition to determining the parameter estimates, Nonlin
can be directed to generate an output file with
predicted values and residuals. It can also plot the
data observations and the computed function. Although
designed for regression analysis, Nonlin can also be
used to find the root (zero point) or minimum absolute
value of a nonlinear expression. Nonlin is in use at
many engineering and research centers around the world.
NONLIN -- Nonlinear Regression Program Page 1
INTRODUCTION TO REGRESSION ANALYSIS
The goal of regression analysis is to determine the values of
parameters for a function that cause the function to best fit a
set of data observations that you provide. In linear regression,
the function is a linear (straight line) equation. For example,
if we assume the value of an automobile decreases by a constant
amount each year after its purchase, and for each mile driven,
the following linear function would predict its value (the
dependent variable) as a function of the two independent
variables which are age and miles:
value = price + depage*age + depmiles*miles
where `value', the dependent variable, is the value of the car,
`age' is the age of the car, and `miles' is the number of miles
that the car has been driven.
The regression analysis performed by Nonlin will determine the
best values of the three parameters, `price', the estimated value
when age is 0 (i.e., when the car was new), `depage', the
depreciation that takes place each year, and `depmiles', the
depreciation for each mile driven. The values of `depage' and
`depmiles' will be negative because the car loses value as age
and miles increase.
In a problem such as this car depreciation example, you must
provide a data file containing the values of the dependent and
independent variables for a set of observations. In this example
each observation record would contain three numbers: value, age,
and miles, collected from used car ads for the same model car.
The more observations you provide, the more accurate will be the
estimate of the parameters.
The Nonlin commands to perform this regression are shown below:
VARIABLES VALUE,AGE,MILES
PARAMETERS PRICE,DEPAGE,DEPMILES
FUNCTION VALUE = PRICE + DEPAGE*AGE + DEPMILES*MILES
DATA
(data values go here)
Once the values of the parameters are determined by Nonlin, you
can use the formula to predict the value of a car based on its
age and miles driven. For example, if Nonlin computed a value of
16000 for price, -1000 for depage, and -0.15 for depmiles, then
the function
value = 16000 - 1000*age - 0.15*miles
NONLIN -- Nonlinear Regression Program Page 2
could be used to estimate the value of a car with a known age and
number of miles.
If a perfect fit existed between the function and the actual
data, the actual value of each car in your data file would
exactly equal the predicted value. Typically, however, this is
not the case, and the difference between the actual value of the
dependent variable and its predicted value for a particular
observation is the error of the estimate which is known as the
"deviation" or "residual". The goal of regression analysis is to
determine the values of the parameters which minimize the sum of
the squared residual values for the set of observations. This is
known as a "least squares" regression fit.
INTRODUCTION TO NONLIN
Nonlin is a very powerful regression analysis program. Using it
you can perform multivariate, linear, polynomial, and general
nonlinear regression. What this means is that you specify the
form of the function to be fitted to the data, and the function
can include nonlinear terms such as variables raised to powers
and library functions such as log, exponential, sine, etc.
Nonlin uses a state-of-the-art regression algorithm that works as
well, or better, than any you are likely to find in commercial
statistical packages.
As an example of nonlinear regression, consider another
depreciation problem. The value of a used airplane decreases for
each year of its age. Assuming the value of a plane falls by the
same amount each year, a linear function relating value to age
is:
Value = p0 + p1*Age
Where `p0' and `p1' are the parameters whose values are to be
determined. However, it is a well known fact that planes (and
automobiles) lose more value the first year than the second, and
more the second than the third, etc. This means that a linear
(straight line) function cannot accurately model this situation.
A better, nonlinear, function is:
Value = p0 + p1*exp(-p2*Age)
Where the `exp' function is the value of e (2.7182818...) raised
to a power. This type of function is known as "negative
exponential" and is appropriate for modeling a value whose rate
of decrease is proportional to the difference between the value
and some base value. The F33YEAR.NLR example command file fits a
linear function to the value of used airplanes. The F33EXP.NLR
example fits a negative exponential function to the same data.
Run both examples and compare the fitted functions. See F33.NLR
for an example of a multiple regression using three independent
NONLIN -- Nonlinear Regression Program Page 3
variables.
Much of the convenience of Nonlin comes from the fact that you
can enter complicated functions using ordinary algebraic
notation. Examples of functions that can be handled with Nonlin
include:
Linear: Y = p0 + p1*X
Quadratic: Y = p0 + p1*X + p2*X^2
Multivariate: Y = p0 + p1*X + p2*Z + p3*X*Z
Exponential: Y = p0 + p1*exp(X)
Periodic: Y = p0 + p1*sin(p2*X)
Misc: Y = p0 + p1*Y + p2*exp(Y) + p3*sin(Z)
In other words, the function is a general expression involving
one dependent variable (on the left of the equal sign), one or
more independent variables, and one or more parameters whose
values are to be estimated.
Because of its generality, Nonlin can perform all of the
regressions handled by ordinary linear or multivariate regression
programs as well as nonlinear regression. However, in order to
handle nonlinear functions, Nonlin uses an iterative function
optimization algorithm which is slower than the simple linear
regression algorithm and has the potential for not converging to
a solution.
INSTALLING NONLIN
The NONLIN system consists of the following files:
NONLIN.EXE -- The executable program.
NONLIN.DOC -- Documentation file.
NONLIN.FON -- Font file used if you request a plot.
NONLIN.LJF -- HP LaserJet font file used if you print a plot.
*.NLR -- Example command files.
REGISTER.DOC -- Form used to register your use of Nonlin.
To install Nonlin, copy the files into the directory of your
choice. If you do not plan to generated hard copy output for a
LaserJet printer, you may delete the NONLIN.LJF file. If the
NONLIN.FON and NONLIN.LJF files are not in your current
directory, you must place a command of the following form in your
AUTOEXEC.BAT file to tell Nonlin where to look for its font
files:
NONLIN -- Nonlinear Regression Program Page 4
SET NONLIN=directory
Where "directory" is the name of the device and directory where
the files are located. For example, if the files are located in
a directory named NONLIN on the C disk, the following command
could be used:
SET NONLIN=C:\NONLIN
USING NONLIN
Once Nonlin has been installed, it can be started using a DOS
command of the form:
NONLIN command_file [listing_file]
where "command_file" is the name of a file containing Nonlin
commands that control the analysis. The sections that follow
describe these commands. If you specify a command file name
without an extension, ".NLR" is used as the default extension.
If you omit the command file name, Nonlin prints a list of its
commands.
A "listing_file" parameter may be specified on the command line.
If you specify a file name, the output (results) of the
regression analysis are written to this file. If no file name is
specified, the output is written to a file with the same name as
the command_file but with the extension ".LST". If you specify a
listing file name without an extension, ".LST" is provided as the
default extension. Specify NUL for the listing_file if you do
not want to generate an output file.
For example, to process a command file named LINEAR.NLR,
directing output to a file named LINEAR.LST, use the following
command:
NONLIN LINEAR
To do the same analysis, directing the output to a file named
MODEL1.LST, use the following command:
NONLIN LINEAR MODEL1
At this point, I suggest you pause in your reading and try
running a Nonlin example to get a feel for how it works. Several
example files with the extension ".NLR" are provided with the
distribution. LINEAR.NLR is a good one to start with. If you do
not have a graphics monitor, edit the LINEAR.NLR command file
(and other example files) and remove the PLOT command.
NONLIN -- Nonlinear Regression Program Page 5
FUNCTION SPECIFICATION
Much of the power of Nonlin comes from its ability to estimate
the value of parameters that are part of complicated functions
that you enter in ordinary algebraic form. This section explains
the arithmetic operators and built in functions that are used to
specify a function.
Arithmetic Operators
The following arithmetic operators may be used in expressions:
+ addition
- subtraction or unary minus
* multiplication
/ division
** or ^ exponentiation
Exponentiation has the highest precedence, followed by
multiplication and division, and then addition and subtraction.
Parentheses may be used to group terms.
As a convenience, Nonlin allows you to omit the multiplication
operator between a numeric constant and a following variable,
parameter, or function. For example, the expressions "2pi", and
"2 pi" are equivalent to "2*pi". Similarly, "5X" is equivalent
to "5*X". However, if you specify a number before the letter
"E", it will be taken as the exponential form of a number (see
below) rather than the number times the constant e (base of
natural logarithms).
Numeric Constants
Numeric constants may be written in their natural form (1, 0,
1.5, .0003, etc.) or in exponential form, n.nnnEppp, where n.nnn
is the base value and ppp is the power of ten by which the base
is multiplied. For example, the number 1.5E4 is equivalent to
15000. All numbers are treated as "floating point" values,
regardless of whether a decimal point is specified or not. As a
convenience for entering time values, if a value contains one or
more colons, the portion to the left of the colon is multiplied
by 60. For example, 1:00 is equivalent to 60; 1:00:00 is
equivalent to 3600.
Symbolic Constants
You can use the CONSTANT command to associate symbolic names with
constant numeric values. When you use the symbolic name in the
function the numeric value is substituted for the symbolic name.
NONLIN -- Nonlinear Regression Program Page 6
Built-in Constants
There are two built-in numeric constants that may be specified
using symbolic names. The symbolic name "PI" is equivalent to
the value of pi, 3.14159... Similarly, the symbolic constant "E"
is equivalent to the base of natural logarithms, e, 2.7182818...
You may write PI and E using either upper or lower case.
Built in Functions
The following functions are built into Nonlin and may be used in
expressions:
ABS(x) -- Absolute value of x.
ACOS(x) -- Arc cosine of x. Angles are measured in radians.
ASIN(x) -- Arc sine of x. Angles are measured in radians.
ATAN(x) -- Arc tangent of x. Angles are measured in radians.
BETAI(x,a,b) -- Incomplete beta function: Ix(a,b). The
incomplete beta function can be used to compute a variety of
statistical functions. For example, the probability of
Student's t with `df' degrees of freedom can be computed
with BETAI(df/(df+t^2),.5*df,.5). The probability of the F
statistic with df1 and df2 degrees of freedom can be
computed with 2*BETAI(df2/(df2+df1*f),.5*df2,.5*df1).
COS(x) -- Cosine of x. Angles are measured in radians.
COSH(x) -- Hyperbolic cosine of x.
COT(x) -- Cotangent of x. (COT(x) = 1/TAN(x)). Angle in radians.
CSC(X) -- Cosecant of x. (CSC(x) = 1/SIN(x)). Angle in radians.
DEG(x) -- Converts an angle, x, measured in radians to the
equivalent number of degrees.
EI1(alpha,phi) -- Elliptic integral of the first kind. Computes
the integral from 0 to phi radians of the function
d.phi/sqrt(1-k**2*sin(phi)**2), where k = sin(alpha). alpha
and phi must be in the range 0 to pi/2.
EI2(alpha,phi) -- Elliptic integral of the second kind. Computes
the integral from 0 to phi radians of the function
sqrt(1-k**2*sin(phi)**2)*d.phi, where k = sin(alpha). alpha
and phi must be in the range 0 to pi/2.
NONLIN -- Nonlinear Regression Program Page 7
EIC1(alpha) -- Complete elliptic integral of the first kind.
Computes the integral from 0 to pi/2 radians of the function
d.phi/sqrt(1-k**2*sin(phi)**2), where k = sin(alpha). alpha
must be in the range 0 to (less than) pi/2.
EIC2(alpha) -- Complete elliptic integral of the second kind.
Computes the integral from 0 to pi/2 radians of the function
sqrt(1-k**2*sin(phi)**2)*d.phi, where k = sin(alpha). alpha
must be in the range 0 to pi/2.
ERF(x) -- Standard error function of x.
EXP(x) -- e (base of natural logarithms) raised to the x power.
FAC(x) -- x factorial (x!). Note, the FAC function is computed
using the GAMMA function (FAC(x)=GAMMA(x+1)) so non-integer
argument values may be computed.
GAMMA(x) -- Gamma function. Note, GAMMA(x+1) = x! (x factorial).
GAMMAI(x) -- Reciprocal of GAMMA function (GAMMAI(x) =
1/GAMMA(x)).
GAMMALN(x) -- Log (base e) of the GAMMA function.
HAV(x) -- Haversine of x. (HAV(x) = (1-COS(x))/2). Angle in
radians.
J0(x) -- Bessel function of the first kind, order zero.
J1(x) -- Bessel function of the first kind, order one.
JN(n,x) -- Bessel function of the first kind, order n.
LOG(x) -- Natural logarithm of x.
LOG10(x) -- Base 10 logarithm of x.
LOG2(x) -- Base 2 logarithm of x.
MAX(x1,x2) -- Maximum value of x1 or x2.
MIN(x1,x2) -- Minimum value of x1 or x2.
NORMAL(x) -- Normal probability distribution of x. X is in units
of standard deviations from the mean. See also the NPD
function. NORMAL(x) = NPD(x,0,1);
NPD(x,mean,std) -- Normal probability distribution of x with
specified mean and standard deviation. X is in units of
standard deviations from the mean.
NONLIN -- Nonlinear Regression Program Page 8
PAREA(x) -- Area under the normal probability distribution curve
from -infinity to x. (i.e., integral from -infinity to x of
NORMAL(x)).
PULSE(a,x,b) -- Pulse function. If the value of x is less than a
or greater than b, the value of the function is 0. If x is
greater than or equal to a and less than or equal to b, the
value of the function is 1. In other words, it is 1 for the
domain (a,b) and zero elsewhere. If you need a function
that is zero in the domain (a,b) and 1 elsewhere, use the
expression (1-PULSE(a,x,b)).
RAD(x) -- Converts an angle measured in degrees to the equivalent
number of radians.
SEC(x) -- Secant of x. (SEC(x) = 1/COS(x)). Angle in radians.
SEL(a1,a2,v1,v2) -- If a1 is less than a2 then the value of the
function is v1. If a1 is greater than or equal to a2, then
the value of the function is v2.
SIN(x) -- Sine of x. Angles are measured in radians.
SINH(x) -- Hyperbolic sine of x.
SQRT(x) -- Square root of x.
STEP(a,x) -- Step function. If x is less than a, the value of
the function is 0. If x is greater than or equal to a, the
value of the function is 1. If you need a function which is
1 up to a certain value and then 0 beyond that value, use
the expression STEP(x,a). See PIECE.NLR for an example of
this function.
T(n,x) -- Chebyshev polynomial of order n.
TAN(x) -- Tangent of x. Angles are measured in radians.
TANH(x) -- Hyperbolic tangent of x.
Y0(x) -- Bessel function of the second kind, order zero.
Y1(x) -- Bessel function of the second kind, order one.
YN(n,x) -- Bessel function of the second kind, order n.
NONLIN -- Nonlinear Regression Program Page 9
NONLIN COMMAND FILES
The commands described in this section are placed in a command
file. When you start Nonlin, you specify the name of the command
file as a parameter on the command line. For example, if the
command file name is CAR.NLR, the following command would cause
Nonlin to execute the commands in the command file:
NONLIN CAR.NLR
If you do not specify a file name extension for the command file,
".NLR" is used by default. The output of the regression for this
example would be written to a file named CAR.LST. Command files
can be created using a text editor such as EDIT-32, EDLIN, the
DOS version 5 EDIT program, or any other editor or word processor
that is capable of creating an ascii text file without formatting
codes.
Comments may be placed in command files by preceding the comment
with an exclamation point. Entire lines may be used for comments
and comments can be placed at the end of commands.
Command lines can be continued by placing a semicolon character
as the last non-blank character on the line (a comment may follow
the semicolon) and then continuing the command on the following
line(s).
Every command file must contain the following commands:
VARIABLES, PARAMETERS, FUNCTION, and DATA. The DATA statement
introduces the data for the analysis and must be the last command
in the file (data records may follow it). Other, optional,
commands may be interspersed in the command file.
The following is an example of a complete command file:
VARIABLES VALUE,AGE,MILES
PARAMETERS BASE,DEPAGE,DEPMILES
FUNCTION VALUE = BASE + DEPAGE*AGE + DEPMILES*MILES
DATA
(data records follow)
NONLIN COMMANDS
The following is a list of the valid Nonlin commands that can be
placed in a Nonlin command file. Command keywords may be
abbreviated to the first three letters except for CONSTANT,
CONSTRAIN, and CONFIDENCE which require six letters. Nonlin
commands are not case sensitive.
NONLIN -- Nonlinear Regression Program Page 10
TITLE string (optional) -- Specifies a title line that is printed
with the results of the analysis.
VARIABLES var1,var2,... (required) -- Specifies the names of the
variables that will be used in the function. The dependent
variable and the independent variables must be specified.
The order of the variable names must match the order of the
data values on each observation record (the dependent
variable may come before or after the independent
variables). You may define more variables than you actually
use in the function specification. A maximum of 12
variables may be specified. The length of a variable name
is limited to 10 characters. Capitalize the variable names
as you want them displayed in the results.
You may specify all of the variables on a single command
line (which may be continued), or you may use multiple
VARIABLES commands. If you use multiple commands, the order
in which they appear in the command file must match the
order of the variable values on each observation record.
The VARIABLES command must precede the FUNCTION command.
See F33.NLR for an example of a multiple regression using
three independent variables.
PARAMETERS param1[=initial1],param2[=initial2],... (required) --
Specifies the names of the parameters whose values are to be
determined by Nonlin. Nonlin is capable of handling up to
12 parameters. The parameter names may not exceed 10
characters in length. Do not specify any parameters that
are not used in the function. The PARAMETERS command must
precede the FUNCTION command.
Optionally, an initial estimate of the parameter value may
be specified by following the parameter name with an equal
sign and the value. If no value is specified, 1 is used by
default. Specifying an initial value that is near the
actual value usually speeds up the operation of Nonlin and
may enable it to successfully converge to a solution. If
Nonlin is unable to converge to a solution, try specifying
different starting values for the parameters. Try to
specify a value that at least has the correct sign as the
expected final value.
The CONSTRAIN command (described below) can be used to limit
the range of values for parameters. The SWEEP command can
be used to perform the regression analysis with a range of
parameter initial values. The CONSTANT command can be used
to define a parameter with a fixed value.
NONLIN -- Nonlinear Regression Program Page 11
CONFIDENCE [percent] (optional) -- Specifies that a confidence
interval is to be printed for each estimated parameter. The
purpose of regression analysis is to determine the best
estimate of parameter values. However, as with most
statistical calculations, the values determined are
estimates of the true values. The CONFIDENCE command causes
Nonlin to print a table showing the range of possible values
for each parameter given a specified confidence value. The
"percent" parameter spcifies the probability that that the
actual value of the parameter is within the confidence
interval to be computed. For example, the command
CONFIDENCE 95
specifies that the confidence interval(s) are to be computed
such that there is a 95 percent probability that the actual
values of the parameters are within the intervals (or that
there is a 5 percent chance that the parameters are outside
the intervals). The "percent" parameter may range from 50
to 99.999. If the CONFIDENCE command is used without
specifiying a percent value, 90 is used by default.
CONSTANT parameter=value (optional) -- Specifies the name of a
symbolic constant and associates a numeric value. You may
then use the symbolic name in the function and the
corresponding constant numeric value will be substituted.
This is useful when you are trying out different models and
want to easily be able to change a constant value for each
run. The CONSTANT commands must precede the FUNCTION
command. The following is an example of a symbolic constant
named "Roomtemp" that causes the value 73 to be substituted
in the function:
Variable Time ! Cooling time in seconds
Variable Temp ! Temperature of object
Constant Roomtemp = 73 ! Ambient temperature
Parameter InitTemp ! Initial temperature
Parameter Coolrate ! Cooling rate factor
Function Temp = Roomtemp + InitTemp * exp(-Coolrate * Time)
CONSTRAIN parameter=lowvalue,highvalue (optional) -- Specifies a
lower and upper limit on the range of a parameter value.
During the solution process, Nonlin may allow a parameter's
value to temporarily move in a direction away from its final
value. With some functions it may be necessary to constrain
the parameter's value so that it does not go negative (e.g.,
if the function takes the square root of the parameter), or
zero (if the parameter is in a denominator). If a parameter
is tightly constrained, Nonlin may report "singular
convergence" because it is unable to converge to an optimum
value of the parameter; however, the estimated values of
other parameters may be useful.
NONLIN -- Nonlinear Regression Program Page 12
Only a single parameter and its associated limits may be
specified on each CONSTRAIN command, but you may use
multiple CONSTRAIN commands. The PARAMETERS command must
precede the CONSTRAIN command. Use the CONSTANT command if
you wish to define a parameter with a fixed value.
The parameter value is allowed to range from `lowvalue' to
`highvalue'. If you want to prevent a parameter value from
going to zero, you must specify a value greater than zero
for the low value (specifying zero would allow it to reach,
but not go below, zero). For example, the following command
constrains the value of `age' to be greater than zero and
less than or equal to 100:
CONSTRAIN age = .0001,100
See the COOLING.NLR, F33EXP.NLR, and POWER.NLR files for
examples of the CONSTRAIN command.
COVARIANCE (optional) -- Causes the variance-covariance matrix
for the parameters to be printed.
SWEEP parameter=lowvalue,highvalue,stepsize (optional) --
Specifies that the regression analysis is to be performed
repeatedly with a set of starting values for the parameter.
The first analysis is performed with the parameter having
the `lowvalue'; the value of `stepsize' is then added to the
parameter's initial value and the analysis is performed
again. The process is repeated until the value of the
parameter reaches `highvalue'.
Each time the analysis is performed the value of the
residual sum of squares is compared with the best previous
result. The estimated values of the parameters for the best
starting value are saved and used for the final analysis and
report.
Only one parameter may be specified on each SWEEP command,
but you may have as many SWEEP commands as there are
parameters. The number of regression analyses performed
will be equal to the product of the number of parameter
values for each SWEEP command.
The SWEEP command is useful when you are trying to fit a
complicated function that may have "local minimum" values
other than the "global minimum". Periodic functions (sin,
cos, etc.) are especially troublesome.
See the SINE.NLR command file for an example of the SWEEP
command.
NONLIN -- Nonlinear Regression Program Page 13
FUNCTION depvar = function (required) -- Specifies the form of
the function whose parameters are to be determined. The
dependent variable must be the only thing to the left of the
equal sign. The expression to the right of the equal sign
may contain variables, parameters, constants, operators, and
library functions such as sqrt, sin, exp, etc. The
VARIABLES and PARAMETERS commands must have appeared in the
command file before the FUNCTION command, and all variables
and parameters used in the function must have been specified
on those commands. Some example FUNCTION commands are show
below:
FUNCTION Y = P0 + P1*X
FUNCTION DISTANCE = .5 * ACCEL * TIME^2
FUNCTION VALUE = PRICE + YRDEP*AGE + MILEDEP*MILES
FUNCTION POPULATN = BASE * GROWRATE * EXP(TIME)
TOLERANCE value (optional, default=1E-10) -- Specifies the
tolerance factor that is used to determine when the
algorithm has converged to a solution. Reducing the
tolerance value may produce a slightly more accurate result
but will increase the number of iterations and the running
time.
ITERATIONS value (optional, default=50) -- Specifies the maximum
number of iterations that should be attempted by the
algorithm. If the solution does not converge to the limit
specified by the TOLERANCE command (or to the default
tolerance) before the maximum number of iterations is
reached, the process is stopped and the results are printed.
Failure to converge before the specified number of
iterations could be caused by one of three things:
1. The maximum allowed number of iterations may be too
small. Try using an ITERATIONS command with a larger value.
2. The tolerance factor may be too small. Even a properly
converging solution will eventually "level off" or oscillate
around a good, but non-zero, sum of squares value. Try
using the TOLERANCE command to increase the tolerance value.
3. The function may not be converging. Try specifying
better (or at least different) starting values for the
parameters on the PARAMETERS command. Consider using the
SWEEP command to specify a range of parameter starting
values.
NONLIN -- Nonlinear Regression Program Page 14
REGISTER (optional) -- The REGISTER command suppresses the
copyright and registration message that is normally printed
as part of a Nonlin report. The use of this command is a
reminder that you should register your use of Nonlin. Note,
if you find Nonlin to be useful, educational, or
entertaining you are expected to register your use so that
the author can be justly compensated and that development of
the program can continue. Use the form in REGISTER.DOC to
register your use.
OUTPUT [TO file] var1,var2,... (optional) -- Specifies that after
the analysis is completed, data values are to be printed or
written to a file. If the "TO file" portion of the command
is specified, the output is written to the specified file.
If this portion of the command is omitted, the output values
are printed along with the results. If a file name is
specified without an extension, ".OUT" is used by default.
The list of variable names determines which variables are
written to the file and the order in which the values appear
in each output record. Any variable previously declared on
a VARIABLES command may be specified. In addition, the
folowing special variable names may appear in the output
list:
$OBS -- The observation record number, starting at 1 and
increasing by 1.
$PREDICTED -- The predicted value for the dependent variable
for the observation, given the independent variable values
and the parameters as calculated by the analysis.
$RESIDUAL -- The difference between the actual value of the
dependent variable and its predicted value.
Examples of OUTPUT commands are shown below:
OUTPUT AGE,MILES,VALUE,$PREDICTED,$RESIDUAL
OUTPUT TO GROWTH.DAT $OBS,TIME,POPULATN,$PREDICTED
POUTPUT file (optional) -- The POUTPUT command specifies that
Nonlin is to write the final estimated values of the
parameters to a file. Each parameter value is written to a
separate line of the file. This command is useful to create
a file of estimated parameter values to be fed into another
analysis program.
PLOT [options] (optional) -- Display a plot of the calculated
function and the data observations. The PLOT command can
only handle a single independent variable (multiple
independent variables would require an n-dimensional surface
plot); however, there is no restriction on the number of
NONLIN -- Nonlinear Regression Program Page 15
parameters being estimated.
You must have a CGA, EGA, or VGA monitor to use the PLOT
command, and the NONLIN.FON font file must be in the current
directory or in a directory specified by the NONLIN
environment variable. In the plot, the data values you
provided are shown as blue X's and the function fitted to
the data by Nonlin is shown as a solid green line. Press
Return to proceed with the analysis after you have finished
looking at the plot.
The following options may be specified on the PLOT command:
GRID -- display grid lines to make it easier to estimate
values.
RESIDUAL -- draw vertical lines from each observed data
point to the corresponding point on the calculated function
line. These lines represent the "residual" value that
Nonlin is attempting to minimize. See also the descriptions
of the RPLOT and NPLOT commands.
ITERATION -- draw a plot for each iteration of the
regression analysis. Normally, the plot is drawn after the
analysis has converged to a solution; you may use the
ITERATION option to observe the function during each
iteration of the analysis as it converges to fit the data.
VALUES -- use in conjunction with the ITERATION option to
cause the current parameter values to be displayed before
the plot for the current iteration.
PRINT -- print a copy of the plot on an HP LaserJet printer.
Nonlin writes the plot to the PRN device which much be
attached to an HP Series II or Series III printer. The
NONLIN.LJF font file must be in the current directory or in
a directory specified by the NONLIN environment variable.
NOPAUSE -- do not pause after the plot is displayed.
Normally, Nonlin pauses after displaying a plot to allow you
time to examine it; you press Enter to continue execution
once you have finished looking at the plot. The NOPAUSE
option causes Nonlin to continue with execution without
pausing after the plot is displayed. This is useful in
conjunction with the PRINT option when Nonlin is run in a
batch file and you want to generate a hardcopy plot but not
pause after the screen display.
The option keywords may be abbrievated to their first
letter. If more than one option is specified, separate them
with commas. For example, to produce a plot with both grid
lines and residual lines, use the following command:
NONLIN -- Nonlinear Regression Program Page 16
PLOT GRID,RESIDUAL
RPLOT [options] (optional) -- Display a plot of the residual
values. A "residual" value (or error deviation) is the
difference between an actual value of the dependent variable
for an observation and the predicted value based on the
function fitted by the regression analysis. If the
calculated function exactly predicted the actual observation
values, all of the residual values would be zero. However,
this is usually not the case and the residual values show
where, and by how much, the fitted function fails to predict
the actual observations.
The RPLOT command causes Nonlin to display a plot showing
the residual values on the vertical (Y) axis and the
independent variable values on the horizontal (X) axis.
However, if there is more than one independent variable,
Nonlin displays the residual values on the vertical (Y) axis
and the dependent variable values on the horizontal (X)
axis. The plot title indicates if the dependent variable
was used for the X axis.
A residual plot is very useful for determining if the form
of the function being fitted is appropriate for the data
values. If the residual values are randomly distributed in
positive and negative directions then the form (shape) of
the fitted function is probably appropriate for the data and
the deviations are due to random measurement errors. If,
however, the residuals show a systematic pattern such as a
periodic cycle, then the function may not be appropriate for
the data values. See the discussion of the Durbin-Watson
statistic for additional information about autocorrelated
residual values. The PLOT, RPLOT, and NPLOT commands may be
used in the same command file. Press Return to proceed with
the analysis after you have finished looking at the plot.
The following options may be specified on the RPLOT command:
GRID -- display grid lines to make it easier to estimate
values.
PRINT -- print a copy of the plot on an HP LaserJet printer.
Nonlin writes the plot to the PRN device which much be
attached to an HP Series II or Series III printer.
NOPAUSE -- do not pause after the plot is displayed.
Normally, Nonlin pauses after displaying a plot to allow you
time to examine it; you press Enter to continue execution
once you have finished looking at the plot. The NOPAUSE
option causes Nonlin to continue with execution without
pausing after the plot is displayed.
NONLIN -- Nonlinear Regression Program Page 17
The option keywords may be abbrievated to their first
letter. If more than one option is specified, separate them
with commas.
NPLOT [options] (optional) -- Display a normal probability plot
of the residual values. In this plot, the actual value of
each residual is plotted on the vertical (Y) axis and the
expected value of the residual, assuming the residuals are
normally distributed, is plotted on the horizontal (X) axis.
If the residuals are normally distributed, the resulting
plot will be a straight line passing through the origin with
a slope of 1 (i.e., the actual value of each residual should
equal the expected value from the normal distribution). If
the residuals are not normally distributed, the plot will
deviate from a straight line.
This plot also computes the correlation between the actual
residual values and their expected values and displays the
correlation coefficient in the title line "(r=n.nn)". If
the residual values are normally distributed, the
correlation should be close to 1.00. A correlation value
less than 0.94 suggests that the residuals are not normally
distributed.
The NPLOT command may be used even if there is more than one
independent variable. The PLOT, RPLOT, and NPLOT commands
may be used in the same command file. Press Return to
proceed with the analysis after you have finished looking at
the plot.
The following options may be specified on the NPLOT command:
GRID -- display grid lines to make it easier to estimate
values.
PRINT -- print a copy of the plot on an HP LaserJet printer.
Nonlin writes the plot to the PRN device which much be
attached to an HP Series II or Series III printer.
NOPAUSE -- do not pause after the plot is displayed.
The option keywords may be abbrievated to their first
letter. If more than one option is specified, separate them
with commas.
DOMAIN lowvalue,highvalue (optional) -- Specifies the domain over
which the plot is to be generated. If the DOMAIN statement
is omitted, the domain of the independent variable is used
for the plot. The DOMAIN statement can be used to generate
a plot of the fitted function extrapolated over the
specified domain. You can also use the DOMAIN command to
restrict the domain and "zero in" on a particular range of
the function. The DOMAIN command only affects the PLOT
NONLIN -- Nonlinear Regression Program Page 18
command; it does not affect the regresson calculation or the
RPLOT or NPLOT commands.
DATA [file] (required) -- Specifies the name of the file
containing the data records, or introduces the data records
which follow the command. If a file name is specified on
the DATA command, the file is opened, its data records are
read, and the regression analysis is performed. If a file
name is specified without an extension, ".DAT" is used by
default.
If no file name is specified on the DATA command, the data
records must immediately follow the DATA command in the
command file.
Each data record must contain at least as many data values
as the number of variables specified on the VARIABLES
command(s). The order of the variables as specified on the
VARIABLES command must match the order of the values in each
observation. Any data values beyond those required for the
specified variables are ignored. Each observation must
begin on a new line.
The data values must be separated by one or more spaces
and/or a comma. Data values may contain decimal points and
may be expressed in exponential notation (i.e., n.nnnnEppp).
As a convenience for entering time values, if a value
contains one or more colons, the portion to the left of the
colon is multiplied by 60. For example, 1:00 is equivalent
to 60; 1:00:00 is equivalent to 3600.
You may continue data lines by specifying a semicolon as the
last non-blank character on a record and then placing the
continuation value on the following line(s).
The DATA command must be the last command in the command
file. If no file name is specified on the DATA command, the
data records must immediately follow the DATA command in the
command file.
The following is an example of a complete command file
including data records:
VARIABLES AGE,MILES,VALUE
PARAMETERS BASE,DEPAGE,DEPMILES
FUNCTION VALUE = BASE + DEPAGE*AGE + DEPMILES*MILES
DATA
2 10000 13000
4 42000 9000
1 7000 17000
6 52000 6000
5 48000 8000
NONLIN -- Nonlinear Regression Program Page 19
If the data records had been placed in a separate file named
CAR.DAT, the DATA statement would be changed to
"DATA CAR.DAT".
UNDERSTANDING THE RESULTS
Nonlin prints a variety of statistics at the end of each
analysis. For each variable, Nonlin lists the minimum value, the
maximum value, the mean value, and the standard deviation. You
should confirm that these values are within the ranges you
expect.
For each parameter, Nonlin displays the initial parameter
estimate (which you specified on the PARAMETER command, or 1 by
default), the final (maximum likelihood) estimate, the standard
error of the estimated parameter value, the "t" statistic
comparing the estimated parameter value with zero, and the
significance of the t statistic.
The final estimate parameter values are the results of the
analysis. By substituting these values in the equation you
specified to be fitted to the data, you will have a function that
can be used to predict the value of the dependent variable based
on a set of values for the independent variables. For example,
if the equation being fitted is
y = p0 + p1*x
and the final estimates are 1.5 for p0 and 3 for p1, then the
equation
y = 1.5 + 3*x
is the best equation of this form that will predict the value of
y based on the value of x.
The "t" statistic is computed by dividing the estimated value of
the parameter by its standard error. This statistic is a measure
of the likelihood that the actual value of the parameter is not
zero. The larger the absolute value of t, the less likely that
the actual value of the parameter could be zero.
The "Prob(t)" is the probability that the actual value of the
parameter could be zero. The smaller the value of Prob(t), the
more significant the parameter. For example, assume the
estimated value of a parameter is 1.0 and its standard error is
0.7. Then the t value would be 1.43 (1.0/0.7). If the computed
Prob(t) value was 0.05 then this indicates that there is only a
0.05 (5%) chance that the actual value of the parameter could be
zero. If Prob(t) was 0.001 this indicates there is only 1 chance
in 1000 that the parameter could be zero. If Prob(t) was 0.92
this indicates that there is a 92% probability that the actual
NONLIN -- Nonlinear Regression Program Page 20
value of the parameter could be zero; this implies that the term
of the regression equation containing the parameter can be
eliminated without significantly affecting the accuracy of the
regression. The t statistic probability is computed using a
two-sided test. The CONFIDENCE command can be used to cause
Nonlin to print confidence intervals for parameter values. The
SQUARE.NLR example regression includes an extraneous parameter
(p0) whose estimated value is much smaller than its standard
error; the Prob(t) value is 0.99982 indicating that there is a
high probability that the value is zero.
In addition to the variable and parameter values, Nonlin displays
several statistics that indicate how well the equation fits the
data. The "Final sum of squared deviations" is the sum of the
squared differences between the actual value of the dependent
variable for each observation and the value predicted by the
function, using the final parameter estimates.
The "Average deviation" is the average over all observations of
the absolute value of the difference between the actual value of
the dependent variable and its predicted value.
The "Maximum deviation for any observation" is the maximum
difference (ignoring sign) between the actual and predicted value
of the dependent variable for any observation.
The "Proportion of variance explained (R^2)" indicates how much
better the function predicts the dependent variable than just
using the mean value of the dependent variable. This is also
known as the "coefficient of multiple determination." It is
computed as follows: Suppose that we did not fit an equation to
the data and ignored all information about the independent
variables in each observation. Then, the best prediction for the
dependent variable value for any observation would be the mean
value of the dependent variable over all observations. The
"variance" is the sum of the squared differences between the mean
value and the value of the dependent variable for each
observation. Now, if we use our fitted function to predict the
value of the dependent variable, rather than using the mean
value, a second kind of variance can be computed by taking the
sum of the squared difference between the value of the dependent
variable predicted by the function and the actual value.
Hopefully, the variance computed by using the values predicted by
the function is better (i.e., a smaller value) than the variance
computed using the mean value. The "Proportion of variance
explained" is computed as 1 - (variance using predicted value /
variance using mean). If the function perfectly predicts the
observed data, the value of this statistic will be 1.00 (100%).
If the function does no better a job of predicting the dependent
variable than using the mean, the value will be 0.00.
The "adjusted coefficient of multiple determination (Ra^2)" is an
R^2 statistic adjusted for the number of parameters in the
NONLIN -- Nonlinear Regression Program Page 21
equation and the number of data observations. It is a more
conservative estimate of the percent of variance explained,
especially when the sample size is small compared to the number
of parameters. It is computed using the formula:
Ra^2 = 1 - (n-1)/(n-p) * (1-R^2)
where `n' is the number of observations, `p' is the number of
parameters, and `R^2' is the unadjusted coefficient of multiple
determination.
The "Durbin-Watson test for autocorrelation" is a statistic that
indicates the likelihood that the deviation (error) values for
the regression have a first-order autoregression component. The
regression models assume that the error deviations are
uncorrelated. In business and economics, many regression
applications involve time series data. If a non-periodic
function, such as a straight line, is fitted to periodic data the
deviations have a periodic form and are positively correlated
over time; these deviations are said to be "autocorrelated" or
"serially correlated." Autocorrelated deviations may also
indicate that the form (shape) of the function being fitted is
inappropriate for the data values (e.g., a linear equation fitted
to quadratic data).
If the deviations are autocorrelated, there may be a number of
consequences for the computed results: 1) The estimated
regression coefficients no longer have the minimum variance
property; 2) the mean square error (MSE) may seriously
underestimate the variance of the error terms; 3) the computed
standard error of the estimated parameter values may
underestimate the true standard error, in which case the t values
and confidence intervals may be incorrect. Note that if an
appropriate periodic function is fitted to periodic data, the
deviations from the regression will be uncorrelated because the
cycle of the data values is accounted for by the fitted function.
Small values of the Durbin-Watson statistic indicate the presence
of autocorrelation. Consult significance tables in a good
statistics book for exact interpretations; however, a value less
than 0.80 usually indicates that autocorrelation is likely. If
the Durbin-Watson statistic indicates that the residual values
are autocorrelated, it is recommended that you use the RPLOT
and/or NPLOT commands to display a plot of the residual values.
If an NPLOT command is used to produce a normal probability plot
of the residuals, the correlation between the residuals and their
expected values (assuming they are normally distributed) is
printed in the listing. If the residuals are normally
distributed, the correlation should be close to 1.00. A
correlation less than 0.94 suggests that the residuals are not
normally distributed.
NONLIN -- Nonlinear Regression Program Page 22
An "Analysis of Variance" table provides statistics about the
overall significance of the model being fitted.
THEORY OF OPERATION
Nonlin uses a model/trust-region technique along with an adaptive
choice of the model Hessian. The algorithm is essentially a
combination of Gauss-Newton and Levenberg-Marquardt methods;
however, the adaptive algorithm often works much better than
either of these methods alone.
The basis for the minimization technique used by Nonlin is to
compute the sum of the squared residuals for one set of parameter
values and then slightly alter each parameter value and recompute
the sum of squared residuals to see how the parameter value
change affects the sum of the squared residuals. By dividing the
difference between the original and new sum of squared residual
values by the amount the parameter was altered, Nonlin is able to
determine the approximate partial derivative with respect to the
parameter. This partial derivative is used by Nonlin to decide
how to alter the value of the parameter for the next iteration.
If the function being modeled is well behaved, and the starting
value for the parameter is not too far from the optimum value,
the procedure will eventually converge to the best estimate for
the parameter. This procedure is carried out simultaneously for
all parameters and is, in fact, a minimization problem in
n-dimensional space, where `n' is the number of parameters.
For a much more detailed explanation of the regression algorithm
used by Nonlin see ACM Transactions on Mathematical Software 7,3
(Sept. 1981) "Dennis, J.E., Gay, D.M., and Welsch, R.E. -- An
adaptive nonlinear least-squares algorithm."
CONVERGENCE CRITERION
Nonlin has several convergence criterion that stop the iterative
minimization procedure. The TOLERANCE command can be used to
alter the convergence tolerance value.
Two internal variables are used to determine when convergence has
occurred. RFCTOL has a default value of 1E-10 and can be altered
by use of the TOLERANCE command. AFCTOL has a default value of
1E-20 and is only altered by the TOLERANCE command if the value
specified is less than the default value. In the discussion
which follows the "function value" is half the sum of the squared
residuals computed using the current parameter estimates.
"Relative function convergence" is reported if the predicted
maximum possible function reduction is at most RFCTOL*ABS(F0)
where F0 is the function value at the start of the current
NONLIN -- Nonlinear Regression Program Page 23
iteration, and if the last step attempted achieved no more than
twice the predicted function decrease.
"Absolute function convergence" is reported if the function value
is less than AFCTOL.
HINTS FOR NONLIN USE
Convergence Failures
One of the potential problems that confronts any nonlinear
minimization procedure is non-convergence. Non-convergence is
usually not a problem for regressions using a linear model, but
becomes a more serious consideration when using complicated
nonlinear functions; increasing the number of parameters
aggravates the problem.
Non-convergence can occur in two ways: the solution may diverge
or it may converge to the wrong solution -- a local minimum
rather than the global minimum. Periodic functions, such as sin,
and cos, are particularly prone to convergence problems. For
example, consider a nonlinear regression performed with the
function:
y = offset + amplitude * sin(frequency * x)
where x and y are variables, and offset, amplitude, and frequency
are the parameters whose values are to be determined. If the
starting value for frequency is not reasonably close to the
correct value, the solution may converge to a harmonic (multiple)
or subharmonic (fundamental) value of the frequency. A command
file named SINE.NLR is supplied with the commands and data to
perform this analysis.
The SWEEP command can be very useful in cases like the sine
example. In the SINE.NLR example analysis, the actual value of
the frequency is 3; the function converges to the correct
solution if the starting value is in the range 2.6 to 3.3.
However, this example is quite insensitive to the starting value
of the amplitude parameter. With an actual value of 2, the
correct solution is found with starting values from 1 through
10000. Similarly, the offset parameter, which had an actual
value of 10, was successfully determined with starting values
ranging from 1 to over 50000.
Another example which is sensitive to a parameter starting value
is POWER.NLR which attempts to determine the values of the
parameters p0, p1, and p2 for the function
y = p0 + p1*x^p2
NONLIN -- Nonlinear Regression Program Page 24
(where "x^p2" means x raised to the p2 power). The actual value
of p2 in the example data is 2; the solution converges correctly
if the starting value of p2 is in the range 1.8 to 3.8. As with
the other example, the solution is relatively insensitive to the
starting values of p0 and p1.
Singular Matrix Problems
Another possible problem is that the analysis may stop with the
message "Singular convergence. Mutually dependent parameters?".
This is usually due to one of two things: (1) a redundant
parameter that is co-dependent with another parameter, or (2) a
situation where the value of one parameter "blocks" the effect of
other parameters.
As an example of a redundant parameter, consider the function
y = p0 + p1*p2*x
This is a simple linear equation except there are two parameters,
p1, and p2, which are both factors to the variable x. It should
be clear that there is no unique solution to this problem since
any value of p1 is possible if the right value of p2 is chosen.
Similarly, the function
y = p0 + p1 + p2*x
has no unique solution since either p0 or p1 is redundant.
Similarly, in the equation
y = p0 + p1*exp(x+p2)
either p1 or p2 is redundant.
The second type of singular matrix problem can be illustrated by
the function
y = p0 + p1*x^p2
If, during the solution process, p1 takes on the value 0, then
varying the value of p2 has no effect on the equation and Nonlin
cannot figure out which way to change the value of p2 to move
toward convergence. The solution to this problem is to assign a
starting value that is not zero to p1, and use the CONSTRAIN
command to force p1 to remain non-zero.
NONLIN -- Nonlinear Regression Program Page 25
PERFORMANCE ISSUES
Nonlin is carefully programmed and compiled with an optimizing
compiler for maximum performance. However, Nonlin is a real
"number cruncher," and the nonlinear regression algorithm is
mathematically very elaborate. During each iteration, Nonlin
computes gradients, Jacobians, Hessians, and eigenvalues, and
performs QR and Cholesky matrix decompositions. All calculations
are carried out using double precision (64 bit) floating point.
Nonlin does not require an 80x87 numeric coprocessor, but its
performance is greatly enhanced if one is present. In fact, an
8088 CPU with an 8087 numeric coprocessor can perform regression
analyses faster than a 20 MHz 80386 that does not have a
coprocessor. If you have an 8088 without a coprocessor, be
patient -- Nonlin is probably giving it the workout of its life.
Very long running times can result if you use the SWEEP command
with many starting values. The problem is compounded if you have
multiple SWEEP commands. If you use the SWEEP command to try a
large number of starting parameter values, you can save time by
using the ITERATIONS command to specify a small number of
iterations (such as 5) during the initial attempt to find a
solution. Once a feasible set of starting parameter values has
been determined, remove the SWEEP command, specify the starting
values on the PARAMETERS command, increase the number of
iterations, and rerun the analysis to get the final result.
PROGRAM LIMITS
The following is a summary of the Nonlin program limitations:
Maximum number of variables = 12
Maximum number of parameters = 12
Maximum length of variable or parameter names = 10
The maximum number of data observations that Nonlin can handle
depends on the number of parameters as shown by the table that
follows:
# Parameters Max Observations
1 2019
2 1611
3 1339
4 1144
5 997
6 883
7 791
8 715
9 652
10 599
NONLIN -- Nonlinear Regression Program Page 26
EXAMPLE ANALYSES
A number of example regression analysis files are provided with
your Nonlin distribution. All of the example command files have
the extension ".NLR". Some of the important ones are described
below, others contain comment lines that explain what they do.
LINEAR.NLR -- Simple linear regression with plotted function and
data.
QUAD.NLR -- Fit a quadratic equation. Plot the function and the
data.
ASYMPTOT.NLR -- Fit an asymptotic function Y = 12 - 10/X.
F33.NLR -- Multivariate linear regression (multiple regression).
Calculate the value of a used Beech F33 Bonanza airplane
using a linear model based on its age, the number of hours
on its airframe, and the number of hours on its engine. The
t value and Prob(t) indicate that the number of hours on the
engine (`Engdep' parameter) is not signficant to the
regression model; the other parameters are significant but
airframe hours is less significant than the base price and
age of the plane.
F33YEAR.NLR -- Similar to F33.NLR except the price of the Bonanza
is calculated based on a linear function of only the age.
F33EXP.NLR -- Similar to F33YEAR.NLR except a negative
exponential function is used rather than a linear function.
Compare the fit of this model with that of the F33YEAR.NLR
example.
SINE.NLR -- Fit an equation involving a sin function. The SWEEP
command is used to find a starting point that will converge.
SQUARE.NLR -- Fit a sine series to a square wave. Note in this
example that the 'p0' parameter, which represents the
constant term of the equation, has an estimated value of
9.22715E-006 (very nearly zero) and a standard error of
0.0398754. This yields a t value of nearly zero and Prob(t)
of 0.99982 which means that there is a 99.982% chance that
the actual value of p0 may be zero (it is in fact zero).
This illustrates how you can use the t value and Prob(t) to
identify extraneous parameters.
COOLING.NLR -- Fit an equation involving an exponential function.
If a heated object is allowed to cool, the rate of cooling
at any instant is proportional to the difference between the
object's temperature and the ambient (room) temperature. In
other words, an object cools faster at first, while it is
hot, and the rate of cooling slows down as the temperature
of the object approaches the ambient temperature. The
NONLIN -- Nonlinear Regression Program Page 27
function that relates the object's temperature to time is:
Temperature = Roomtemp+InitTemp*exp(-Coolrate*Time)
Where InitTemp is the number of degrees above room
temperature at time 0, and Coolrate is a factor that depends
on the mass of the object, how well it is insulated, etc.
The exp function is the value of e (2.7182818...) raised to
a power. The COOLING.NLR example determines the parameters
InitTemp and Coolrate to fit an equation of this form to
some data the author collected.
BOIL.NLR -- The boiling point of water decreases as the pressure
in the vessel containing the water decreases. "Clapeyron's
equation" shows that the boiling point is related to
pressure according to the following function:
Temperature = b / log(Pressure/a) - 459.7
Where `Temperature' is in degrees Fahrenheit (the 459.7
constant converts degrees Fahrenheit to degrees Rankine --
relative to absolute zero), `Pressure' is the pressure in
the vessel in pounds per square inch, and `a' and `b' are
parameters whose values are to be determined. The data for
this example was collected by the author's son for a science
project.
MAGNET.NLR -- Fit a function involving an arc tangent and a
variable to the third power. This is an interesting physics
problem. If a magnet is placed due east of a compass, the
deflection of the compass needle from north is equal to the
arc tangent of the ratio of the strength of the magnet's
field relative to the earth's magnetic field. The strength
of the magnet's field at the compass is inversely
proportional to the cube of the distance from the magnet to
the compass. Thus, the function relating these terms is
Deflection = deg(atan(Strength / Distance ^ 3))
The deg function converts an angle in radians to degrees.
In the example, Deflection and Distance are the variables,
and the value of the Strength parameter is determined.
DIODE.NLR -- The current through a diode increases sharply as the
voltage across the diode is increased. An equation that
approximates the current flow as a function of the voltage
is:
I = exp(b*(V-c))
where `I' is the current, `V' is the voltage, and `b', and
`c' are parameters that are to be estimated by the nonlinear
regression.
NONLIN -- Nonlinear Regression Program Page 28
AVLTIME.NLR -- An AVL tree is a balanced binary tree used to
store information in a computer's memory. Because the
entries in an AVL tree are kept in sorted order, and the
tree is kept in a balanced form, it is possible to rapidly
find any entry in the tree. The time required to create an
AVL tree with N entries is approximately equal to:
Time = a + b*N*log2(N)
where `a' is a constant term equal to the overhead involved
in starting and completing a tree creation, and `b' is a
growth coefficient that depends on the speed of the
computer. The log2(N) function is the log base 2 of N (the
number of entries). The AVLTIME.NLR example fits an
equation to a data set that relates the time in seconds
required to create an AVL tree with the number of entries in
the tree.
PIECE.NLR -- Piecewise linear function. Fit a function
consisting of two linear pieces that bend at X=5. When X is
less than 5, the slope of the function is B1. When X is
greater than or equal to 5, the slope is B2. B0 is the Y
value of the function at X=5 (i.e., at the pivot point).
The step(a,x) function returns the value 0 when x is less
than `a'; it returns 1 when x is greater than or equal to
`a'.
NONLIN USE FOR ROOT FINDING AND FUNCTION MINIMIZATION
Although it is designed for nonlinear regression analysis, Nonlin
can also be used to find the root (zero point) or minimum
absolute value of a nonlinear expression. To use Nonlin in this
fashion follow these steps: (1) Do not use any VARIABLE
statements; (2) Use PARAMETER statements to specify the names and
optional starting values for the parameters whose values are to
be determined as the roots or minimum value of the expression;
(3) Use the FUNCTION statement to specify the expression whose
roots or minimum value is to be found; do NOT specify a dependent
variable and equal sign -- specify only the expression that is to
be minimized; (4) Do not include any data records after the DATA
statement; it simply signals the end of the command file and
causes the analysis to begin.
The following is an example command file to find the root of the
expression SIN(X)-LOG(X):
PARAMETER X
FUNCTION SIN(X) - LOG(X)
DATA
NONLIN -- Nonlinear Regression Program Page 29
Notice that the "variable" in the expression, X, is not declared
to be a variable but rather a parameter. This example is
included in the file MINSL.NLR which you can run.
For this type of analysis, Nonlin determines the values of the
parameters that minimize the absolute value of the expression.
If the expression has a zero value (i.e., a root), that value is
found since that is the smallest possible absolute value. If the
expression does not have a zero point, Nonlin determines the
values of the parameters that produce the smallest absolute value
of the expression. For example, the expression 2*x^2-3*x+10 does
not have a root but reaches a minimum value of 8.875 when x is
0.75. The MINPAROB.NLR command file contains this example.
There are a number of cautions that you should keep in mind when
using Nonlin to find roots or minimum values:
1. Nonlin will find only one root or minimum value per analysis.
For example, the expression 9-x^2 has two roots: -3 and +3.
Nonlin will find one of the roots; which one it finds depends
on the starting value specified for X.
2. Nonlin will find only real roots, not complex.
3. If the expression contains a local minimum, Nonlin may find it
rather than the global minimum or root. Of course, if you are
looking for a local minimum in a certain region this could be
considered a feature. For example, the expression
0.5*x^3+5*(x-2)^2+15 has a local minimum at x=1.61 and a root
at x=-13.38. If the starting value of x is less than -8.3 the
root is found; if the starting value is greater than -8.3, the
local minimum is found. If the expression contains only a
single variable, use the Mathplot program to graphically
display the expression and determine a good starting value for
the variable (see the end of this document for additional
information about Mathplot). The SWEEP command can also be
used to try multiple starting values when searching for a
global minimum.
FUNCTION MINIMIZATION EXAMPLES
MINFALL.NLR -- The time taken for an object to slide down a
frictionless guide from position (0,h) to another position
(d,0) (i.e., falling through a distance `h' while moving
horizontally a distance `d') depends on the path that the
object takes as it follows the guide. It turns out that the
path that minimizes the descent time is not a straight line
from (0,h) to (d,0) but rather a curve called a
brachistochrone with a steeper slope near the beginning,
that gives the object a chance to accelerate quickly, and
then a shallower slope further on. Finding the shape of
this curve is a classic problem in the branch of mathematics
called the Calculus of Variations. The MINFALL example
NONLIN -- Nonlinear Regression Program Page 30
solves a simpler case of this problem: the object slides
along a straight guide from (0,1000) to an intermediate
position (px,py), and then along another straight guide from
(px,py) to (1000,0). What point, (px,py), minimizes the
descent time? [Note concerning the answer: The fall time for
the object if it follows a straight guide from (0,1000) to
(1000,0) is 2.0203 seconds; the fall time if it follows the
two straight segments found by MINFALL is 1.8748; the fall
time if it follows the ideal curved brachistochrone is
1.8590. The speed of the object at the end of the fall is
the same regardless of the path taken (conservation of
energy).]
MINFUEL.NLR -- A lunar lander is hovering above the surface of
the moon looking for a suitable landing site. Available
fuel is critical and the desired site is 200 meters away.
How long should the horizontal thruster be fired to start
and stop the motion over the ground? The vertical thruster
must be used continuously to keep the lander from being
pulled to the surface. If too little horizontal thrust is
used the spacecraft will move slowly and much fuel will be
consumed by the vertical thruster counterbalancing the
downward gravitational pull while hovering over the surface.
On the other hand, if the horizontal thruster is fired for a
a long time, the spacecraft will move quickly (minimizing
the hovering time) but excessive fuel will be used during
the horizontal acceleration and deceleration. MINFUEL.NLR
determines how long the thruster should be fired during the
start and stop accelerations such that the total fuel
consumption (start thrust + stop thrust + hover) is
minimized.
ACKNOWLEDGEMENT
The nonlinear regression algorithm used by Nonlin was published
in ACM Transactions on Mathematical Software 7,3 (Sept. 1981)
"Dennis, J.E., Gay, D.M., and Welsch, R.E. -- An adaptive
nonlinear least-squares algorithm."
USE AND DISTRIBUTION OF NONLIN
Nonlin is a "shareware" product. You are welcome to make copies
of this program and pass them on to friends or post this program
on bulletin boards (in its original, unmodified form).
As a shareware product, you are granted a no-cost, trial period
of 30 days during which you may evaluate Nonlin. If you find
Nonlin to be useful, educational, and/or entertaining, and
continue to use it beyond the 30 day trial period, you are
required to compensate the author by sending the registration
form printed at the end of this document (and in REGISTER.DOC)
NONLIN -- Nonlinear Regression Program Page 31
with the appropriate registration fee to help cover the
development and support of Nonlin.
In return for registering, you will be authorized to continue
using Nonlin beyond the trial period and you will receive the
most recent version of the program, a laser-printed, bound
manual, and three months of support via telephone, mail, or
CompuServe. Your registration fee will be refunded if you
encounter a serious bug that cannot be corrected.
The author frequently improves Nonlin and it is likely that the
version you have is not the most recent version. Note, the cost
of registering Nonlin is insignificant compared with what you
would have to pay to purchase a commercial statistical package
with an equivalent regression capability.
You are welcome to contact the author:
Phillip H. Sherrod
4410 Gerald Place
Nashville, TN 37205-3806
615-292-2881 (evenings)
CompuServe: 71333,27
Both the Nonlin program and documentation are copyright (c) 1992
by Phillip H. Sherrod. You are not authorized to modify the
program. "Nonlin" is a trademark.
Disclaimer
Nonlin is provided "as is" without warranty of any kind, either
expressed or implied. This program may contain "bugs" and
inaccuracies, and its results should not be assumed to be correct
unless they are verified by independent means. The author
assumes no responsibility for the use of Nonlin and will not be
responsible for any damage resulting from its use.
NONLIN -- Nonlinear Regression Program Page 32
M A T H P L O T
Mathematical Function Plotting Program
Special Offer
If you like Nonlin, you should check out the Mathplot program by
the same author.
Mathplot allows you to specify complicated mathematical functions
using ordinary algebraic expressions and immediately plot them.
Four types of functions may be specified: cartesian (Y=f(X));
parametric cartesian (Y=f(T) and X=f(T)); polar
(Radius=f(Angle)); and parametric polar (Radius=f(T) and
Angle=f(T)). Up to four functions may be plotted simultaneously.
Scaling is automatic. Options are available to control axis
display and labeling as well as grid lines. Hard copy output may
be generated as well as screen display. Mathplot is an ideal
tool for engineers, scientists, math and science teachers, and
anyone else who needs to quickly visualize mathematical
functions.
SPECIAL OFFER
Registered users of Nonlin can order Mathplot for a special price
of $18. Or, for an even better deal, if you register Nonlin and
order Mathplot at the same time, you can get both for $40.
NONLIN -- Nonlinear Regression Program Page 33
TSX-32 Multi-User Operating System
If you have a need for a multi-user, multi-tasking operating
system, you should look into TSX-32. TSX-32 is a full-featured,
high performance, multi-user operating system for the 386 and 486
that provides both 32-bit and 16-bit program support. With
facilities such as multitasking and multisessions, networking,
virtual memory, background batch queues, data caching, file
access control, real-time, and dial-in support, TSX-32 provides a
solid environment for a wide range of applications.
TSX-32 is not a limited, 16-bit, multi-DOS add-on. Rather, it is
a complete 32-bit operating system which makes full use of the
hardware's potential, including protected mode execution, virtual
memory, and demand paging. TSX-32 sites range from small systems
with 2-3 terminals to large installations with more than 64
terminals on a single 386.
In addition to supporting most popular 16-bit DOS programs,
TSX-32 also provides a 32-bit "flat" address space with both Phar
Lap and DPMI compatible modes of execution.
Since the DOS file structure is standard for TSX-32, you can
directly read and write DOS disks. And, you can run DOS part of
the time and TSX-32 the rest of the time on the same computer.
TSX-32 allows each user to control up to 10 sessions. Programs
can also "fork" subtasks for multi-threaded applications. The
patented Adaptive Scheduling Algorithm provides consistently good
response time under varying conditions.
The TSX-32 network option provides industry standard TCP/IP
networking through Ethernet and serial lines. Programs can
access files on remote machines as easily as on their own
machine. The SET HOST command allows a user on one machine to
log onto another computer in the network. FTP, Telnet, and NFS
are available for interoperability with other systems.
TSX-32 allows simultaneous real-time program execution with
normal time-sharing operations. Real-time programs can connect
to interrupts, access device control ports, and preempt other
operations when necessary. TSX-32 is an ideal process control or
data collection system.
TSX-32 is, quite simply, the best and most powerful operating
system available for the 386 and 486. For additional information
contact:
S&H Computer Systems, Inc.
1027 17th Avenue South
Nashville, TN 37212
615-327-3670 (voice)
615-321-5929 (fax)
NONLIN -- Nonlinear Regression Program Page 34
=====================================================================
Software Order Form
=====================================================================
Name ______________________________________________________
Address ___________________________________________________
City _______________________ State _______ Zip ___________
Telephone _________________________________________________
CompuServe account (optional) _____________________________
Nonlin version ____________________________________________
Bulletin board where you found Nonlin _____________________
Comments __________________________________________________
Check the box below which indicates your order type:
___ I wish to register Nonlin ($25).
___ I wish to order Mathplot ($20).
___ I wish to register Nonlin and order Mathplot ($40).
Add $5 to any amount shown above if the software is being shipped
out of the United States.
In return for registering, you will receive the most recent
version of the program, a laser-printed, bound copy of the
manual, and three months of telephone or CompuServe support.
Your registration fee will be refunded if you find a serious bug
that cannot be corrected.
Distribution disk choice (check one):
3.50" HD (1.4 MB) ______
5.25" HD (1.2 MB) ______
5.25" DD (360 KB) ______
Send this form with the amount indicated to the author:
Phillip H. Sherrod
4410 Gerald Place
Nashville, TN 37205-3806
615-292-2881 (evenings)
CompuServe: 71333,27
